*A Short History of Astronomy*

terrestrial bodies wherever situated, it was probable that at such a distance as that of the moon the acceleration caused by the earth would be much less. Newton assumed as a working hypothesis that the acceleration diminished according to the same law which he had previously arrived at in the case of the sun's action on the planets, that is that the acceleration produced by the earth on any body is inversely proportional to the square of the distance of the body from the centre of the earth.

It may be noticed that a difficulty arises here which did not present itself in the corresponding case of the planets. The distances of the planets from the sun being large compared with the size of the sun, it makes little difference whether the planetary distances are measured from the centre of the sun or from any other point in it. The same is true of the moon and earth; but when we are comparing the action of the earth on the moon with that on a stone situated on or near the ground, it is clearly of the utmost importance to decide whether the distance of the stone is to be measured from the nearest point of the earth, a few feet off, from the centre of the earth, 4000 miles off, or from some other point. Provisionally at any rate Newton decided on measuring from the centre of the earth.

It remained to verify his conjecture in the case of the moon by a numerical calculation; this could easily be done if certain things were known, *viz.* the acceleration of a falling body on the earth, the distance of the surface of the earth from its centre, the distance of the moon, and the time taken by the moon to perform, a revolution round the earth. The first of these was possibly known with fair accuracy; the last was well known; and it was also known that the moon's distance was about 60 times the radius of the earth. How accurately Newton at this time knew the size of the earth is uncertain. Taking moderately accurate figures, the calculation is easily performed. In a month of about 27 days the moon moves about 60 times as far as the distance round the earth; that is she moves about 60 × 24,000 miles in 27 days, which is equivalent to about 3,300 feet per second. The acceleration of the moon is therefore measured by the square of this, divided by the